De Sitter space can be defined as a submanifold of Minkowski space in one higher dimension. Take Minkowski space R^1,n with the standard metric:

De Sitter space is the submanifold described by the hyperboloid

where α is some positive constant with dimensions of length. The metric on de Sitter space is the metric induced from the ambient Minkowski metric. One can check that the induced metric is nondegenerate and has Lorentzian signature. (Note that if one replaces α² with - α² in the above definition, one obtains a hyperboloid of two sheets. The induced metric in this case is positive-definite, and each sheet is a copy of hyperbolic n-space.)

De Sitter space can also be defined as the quotient O(1,n)/O(1,n-1) of two indefinite orthogonal groups, which shows that it is a non-Riemannian symmetric space.

Topologically, de Sitter space is R × S^n-1 (so that that if n ≥ 3 then de Sitter space is simply-connected). Given the standard embedding of the unit (n-1)-sphere in Rⁿ with coordinates y_i one can introduce a new coordinate t so that

Plugging in the subscripted x's into the induced 4D metric, embedding the de Sitter space in the five-dimensional Minkowski space R^1,4, and being careful to use the Leibniz rule in differentials in , we find resulting cross terms there vanish on the sphere and one of the remaining squares of sum hypertrig collapse with to produce -dt², so the metric in these coordinates (t plus some set of coordinates on S^n-1) is given by

where is the standard round metric on the (n-1)-sphere, as concurs reference 3.

Properties
The isometry group of de Sitter space is the Lorentz group O(1,n). The metric therefore then has n(n+1)/2 independent Killing vectors and is maximally symmetric. Every maximally symmetric space has constant curvature. The Riemann curvature tensor of de Sitter is given by

De Sitter space is an Einstein manifold since the Ricci tensor is proportional to the metric:

This means de Sitter space is a vacuum solution of Einstein's equation with cosmological constant given by

The scalar curvature of de Sitter space is given by

For the case n = 4, we have Λ = 3/α² and R = 4Λ = 12/α².

Static coordinates

We can introduce static coordinates for de Sitter as follows:

where z_i gives the standard embedding the (n-2)-sphere in R^n-1. In these coordinates the de Sitter metric takes the form:

Note that there is a cosmological horizon at r = α.

See also

• Anti de Sitter space
• de Sitter universe
• Hyperboloid

Cosmic inflation

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In physical cosmology, cosmic inflation is the idea that the nascent universe passed through a phase of exponential expansion that was driven by a negative-pressure vacuum energy density.^[1]

As a direct consequence of this expansion, all of the observable universe originated in a small causally-connected region. Inflation answers the classic conundrum of the big bang cosmology: why does the universe appear flat, homogeneous and isotropic in accordance with the cosmological principle when one would expect, on the basis of the physics of the big bang, a highly curved, inhomogeneous universe? Inflation also explains the origin of the large-scale structure of the cosmos. Quantum fluctuations in the microscopic inflationary region, magnified to cosmic size, become the seeds for the growth of structure in the universe (see galaxy formation and evolution and structure formation).

Inflation was proposed in January, 1980 by Alan Guth^[2][3] and was given its modern form independently by Andrei Linde,^[4] and by Andreas Albrecht and Paul Steinhardt.^[5]

While the detailed particle physics mechanism responsible for inflation is not known, the basic picture makes a number of predictions that have been confirmed by observational tests. Inflation is thus now considered part of the standard hot big bang cosmology. The hypothetical particle or field thought to be responsible for inflation is called the inflaton.

[edit] Overview

Main article: Metric expansion of space

Inflation suggests that there was a period of exponential expansion in the very early universe. Because in a fast expanding universe, the distance to the cosmological horizon is constant, it is not clear whether such a universe should be called "small" or "large". If the philosophical definition of the universe is restricted to be the observable universe, an inflating universe is small, and only becomes large once inflation has ended and the cosmological horizon is free to expand. If the philosophical position is that the universe is mostly unobservable, then the unobservable portion is expanding exponentially.

[edit] Space expands

To say that space expands exponentially means that two inertial observers are drawn farther apart with time. In stationary coordinates for one observer, a patch of an inflating universe has the following polar metric:

This is just like an inside-out black hole metric - it has a zero in the dt component on a fixed radius sphere called the cosmological horizon. Objects are drawn away from the observer at r=0 towards the cosmological horizon, leading them to fall in after a finite proper time. This means that any inhomogeneities are smoothed out, just as any bumps or matter on the surface of a black hole horizon are swallowed and disappear.

Since the space time metric has no explicit time dependence, once an observer has fallen onto the cosmological horizon, observers closer in take its place. This process of falling outward and replacement points closer in are always steadily replacing points further out - an exponential expansion of space-time.

This steady-state exponentially expanding spacetime is called a de Sitter space, and to sustain it there must be a cosmological constant, a vacuum energy proportional to Λ everywhere. The physical conditions from one moment to the next are stable: the rate of expansion, called the Hubble parameter, is nearly constant. Inflation is often called a period of accelerated expansion because the distance between two fixed observers is increasing at an accelerating rate as they move apart. (but Λ can stay approximately constant see deceleration parameter.)

[edit] Few inhomogeneities remain

Cosmic inflation has the important effect of smoothing out inhomogeneities, anisotropies and the curvature of space. This pushes the universe into a very simple state, in which it is completely dominated by the inflaton field, the source of the cosmological constant, and the only significant inhomogeneities are the tiny quantum fluctuations in the inflaton. Inflation also dilutes exotic heavy particles, such as the magnetic monopoles predicted by many extensions to the Standard Model of particle physics. If the universe was only hot enough to form such particles before a period of inflation, they would not be observed in nature, as they would be so rare that it is quite likely that there are none in the Observable universe. Together, these effects are called the inflationary "no-hair theorem"^[6] by analogy with the no hair theorem for black holes.

The "no-hair" theorem works essentially because the cosmological horizon is no different from a black-hole horizon except for philosophical disagreements about what is on the other side. In terms of the unobservable universe, the interpretation of the no-hair theorem is that the unobservable universe expands by an enormous factor during inflation. In an expanding universe, energy densities generally fall as the volume of the universe increases. For example, the density of ordinary "cold" matter (dust) goes as the inverse of the volume: when linear dimensions double, the energy density goes down by a factor of eight. The energy density in radiation goes down even more rapidly as the universe expands. When linear dimensions are doubled, the energy density in radiation falls by a factor of sixteen. During inflation, the energy density in the inflaton field is roughly constant. However, the energy density in inhomogeneities, curvature, anisotropies and exotic particles is falling, and through sufficient inflation these become negligible. This leaves an empty, flat, and symmetric universe, which is filled with radiation when inflation ends.

[edit] Key requirement

A key requirement is that inflation must continue long enough to produce the present observable universe from a single, small inflationary Hubble volume. This is necessary to ensure that the universe appears flat, homogeneous and isotropic at the largest observable scales. This requirement is generally thought to be satisfied if the universe expanded by a factor of at least 10²⁶ during inflation.^[7]

[edit] Reheating

At the end of inflation, a process called reheating occurs, in which the inflaton particles decay into the radiation that starts the hot big bang. It is not known how long inflation lasted but it is usually thought to be extremely short compared to the age of the universe.

[edit] Motivation

Inflation resolves several problems in the Big Bang cosmology that were pointed out in the 1970s.^[8] These problems arise from the observation that to look like it does today, the universe would have to have started from very finely tuned, or "special" initial conditions at the Big Bang. Inflation attempts to resolve these problems by providing a dynamical mechanism that drives the universe to this special state, thus making a universe like ours much more likely in the context of the Big Bang theory.

[edit] Horizon problem

Main article: Horizon problem

The horizon problem^[9][10][11] is the problem of determining why the universe appears statistically homogeneous and isotropic in accordance with the cosmological principle. For example, molecules in a canister of gas are distributed homogeneously and isotropically because they are in thermal equilibrium: gas throughout the canister has had enough time to interact to dissipate inhomogeneities and anisotropies. The situation is quite different in the big bang model without inflation, because gravitational expansion does not give the early universe enough time to equilibrate. In a big bang with only the matter and radiation known in the Standard Model, two widely separated regions of the observable universe cannot have equilibrated because they move apart from each other faster than the speed of light - thus have never come in to causal contact: in the history of the universe, back to the earliest times, it has not been possible to send a light signal between the two regions. Because they have no interaction, it is difficult to explain why they have the same temperature (are thermally equilibrated). This is because the Hubble radius in a radiation or matter-dominated universe expands much more quickly than physical lengths and so points that are out of communication are coming into communication. Historically, two proposed solutions were the Phoenix universe of Georges Lemaître^[12] and the related oscillatory universe of Richard Chase Tolman,^[13] and the Mixmaster universe of Charles Misner.^[10][14] Lemaître and Tolman proposed that a universe undergoing a number of cycles of contraction and expansion could come into thermal equilibrium. Their models failed, however, because of the buildup of entropy over several cycles. Misner made the (ultimately incorrect) conjecture that the Mixmaster mechanism, which made the universe more chaotic, could lead to statistical homogeneity and isotropy.

[edit] Flatness problem

Main article: Flatness problem

Another problem is the flatness problem (which is sometimes called one of the Dicke coincidences, with the other being the cosmological constant problem).^[15][16] It had been known in the 1960s^{[citation needed]} that the density of matter in the universe was comparable to the critical density necessary for a flat universe (that is, a universe whose large scale geometry is the usual Euclidean geometry, rather than a non-Euclidean hyperbolic or spherical geometry).

Therefore, regardless of the shape of the universe the contribution of spatial curvature to the expansion of the universe could not be much greater than the contribution of matter. But as the universe expands, the curvature redshifts away more slowly than matter and radiation. Extrapolated into the past, this presents a fine-tuning problem because the contribution of curvature to the universe must be exponentially small (sixteen orders of magnitude less than the density of radiation at big bang nucleosynthesis, for example). This problem is exacerbated by recent observations of the cosmic microwave background that have demonstrated that the universe is flat to the accuracy of a few percent.^{[citation needed]}

[edit] Magnetic monopole problem

The magnetic monopole problem (sometimes called the exotic relics problem) is a problem that suggests that if the early universe were very hot, a large number of very heavy, stable magnetic monopoles would be produced. This was a problem with Grand Unified Theories, popular in the 1970s and 1980s, which proposed that at high temperatures (such as in the early universe) the electromagnetic force, strong and weak nuclear forces are not actually fundamental forces but arise due to spontaneous symmetry breaking from a much simpler gauge theory.^[17] These theories predict a number of heavy, stable particles which have not yet been observed in nature. The most notorious is the magnetic monopole, a kind of stable, heavy "knot" in the magnetic field.^[18][19] Monopoles are expected to be copiously produced in Grand Unified Theories at high temperature,^[20][21] and they should have persisted to the present day, to such an extent that they would become the primary constituent of the universe.^[22][23] Not only is that not the case, but all searches for them have so far turned out fruitless, placing stringent limits on the density of relic magnetic monopoles in the universe.^[24] A period of inflation that occurs below the temperature where magnetic monopoles can be produced would offer a possible resolution of this problem: monopoles would be separated from each other as the universe around them expands, potentially lowering their observed density by many orders of magnitude.

[edit] History

[edit] Precursors

In the early days of General Relativity, Albert Einstein introduced the cosmological constant to allow a static solution which was a three dimensional sphere with a uniform density of matter. A little later, Willem de Sitter found a highly symmetric inflating universe, which described a universe with a cosmological constant which is otherwise empty.^[25] Einstein's solution is unstable, and if there are small fluctuations, it eventually turns into de Sitter's.

In the early 1970s Zeldovich noticed the serious flatness and horizon problems of big bang cosmology; before his work, cosmology was presumed to be symmetrical on purely philosophical grounds. In the Soviet Union, this and other considerations led Belinski and Khalatnikov to formulate the mixmaster universe, an analysis of the chaos near a singularity in General Relativity. Starobinsky formulated an early chaotic version of inflation in 1979^[26], which was advanced by Vilenkin and Starobinsky. While this was not as transparent a solution to the cosmological problems as Guth's, it remains a possibility.

In the late 1970s, Sidney Coleman applied the instanton techniques developed by Alexander Polyakov and collaborators to study the fate of the false vacuum in quantum field theory. Like a metastable phase in statistical mechanics--- water below the freezing temperature or above the boiling point--- a quantum field would need to nucleate a large enough bubble of the new vacuum, the new phase, in order to make a transition. Coleman found the most likely decay pathway for vacuum decay and calculated the inverse lifetime per unit volume. He eventually noted that gravitational effects would be significant, but he did not calculate these effects and did not apply the results to cosmology.

In 1978, Zeldovich noted the monopole problem, which was an unambiguous quantitative version of the horizon problem, this time in a fashionable subfield of particle physics, which led to several speculative attempts to resolve it. In 1980, working in the west, Alan Guth realized that false vacuum decay in the early universe would solve the problem.

[edit] Guth, Starobinsky and others

Inflation was proposed in January, 1980 by Alan Guth as a mechanism for resolving these problems.^[2][3] Contemporary with Guth, Alexei Starobinsky argued that quantum corrections to gravity would replace the initial singularity of the universe with an exponentially expanding state.^[27] Demosthenes Kazanas anticipated part of Guth's work by suggesting that exponential expansion could eliminate the particle horizon and perhaps solve the horizon problem,^[28] and Sato suggesting that an exponential expansion could eliminate domain walls (another kind of exotic relic.)^[29] Einhorn and Sato^[30] published a model similar to Guth's and showed that it would resolve the puzzle of the magnetic monopole abundance in Grand Unified Theories. Like Guth, they concluded that such a model not only required fine tuning of the cosmological constant, but also would very likely lead to a much too granular universe, i.e., to large density variations resulting from bubble wall collisions.

[edit] Guth

Guth was the first to assemble a complete picture of how all these initial conditions problems could be solved by an exponentially expanding state.

Guth proposed that as the early universe cooled, it was trapped in a false vacuum with a high energy density, which is much like a cosmological constant. As the very early universe cooled it was trapped in a metastable state (it was supercooled) which it could only decay out of through the process of bubble nucleation via quantum tunneling. Bubbles of true vacuum spontaneously form in the sea of false vacuum and rapidly begin expanding at the speed of light. Guth recognized that this model was problematic because the model did not reheat properly: when the bubbles nucleated, they did not generate any radiation. Radiation could only be generated in collisions between bubble walls. But if inflation lasted long enough to solve the initial conditions problems, collisions between bubbles became exceedingly rare. In any one causal patch, it is likely that only one bubble will nucleate.

[edit] Linde, Albrecht and Steinhardt

The bubble collision problem was solved by Andrei Linde^[4] and independently by Andreas Albrecht and Paul Steinhardt^[5] in a model named new inflation or slow-roll inflation (Guth's model then became known as old inflation). In this model, instead of tunneling out of a false vacuum state, inflation occurred by a scalar field rolling down a potential energy hill. When the field rolls very slowly compared to the expansion of the universe, inflation occurs. However, when the hill becomes steeper, inflation ends and reheating can occur.

[edit] Effects of asymmetries

Eventually, it was shown that new inflation does not produce a perfectly symmetric universe, but that tiny quantum fluctuations in the inflaton are created. These tiny fluctuations form the primordial seeds for all structure created in the later universe. These fluctuations were first calculated by Viatcheslav Mukhanov and G. V. Chibisov in the Soviet Union in analyzing Starobinsky's similar model.^[31][32][33] In the context of inflation, they were worked out independently of the work of Mukhanov and Chibisov at the three-week 1982 Nuffield Workshop on the Very Early Universe at Cambridge University.^[34] The fluctuations were calculated by four groups working separately over the course of the workshop: Stephen Hawking;^[35] Starobinsky;^[36] Guth and So-Young Pi;^[37] and James M. Bardeen, Paul Steinhardt and Michael Turner.^[38]

[edit] Observational status

Inflation is a concrete mechanism for realizing the cosmological principle which is the basis of the standard model of physical cosmology: it accounts for the homogeneity and isotropy of the observable universe. In addition, it accounts for the observed flatness and absence of magnetic monopoles. Since Guth's early work, each of these observations has received further confirmation, most impressively by the detailed observations of the cosmic microwave background made by the Wilkinson Microwave Anisotropy Probe (WMAP) satellite.^[39] This analysis shows that the universe is flat to an accuracy of at least a few percent, and that it is homogeneous and isotropic to a part in 10,000.

In addition, inflation predicts that the structures visible in the universe today formed through the gravitational collapse of perturbations which were formed as quantum mechanical fluctuations in the inflationary epoch. The detailed form of the spectrum of perturbations called a nearly-scale-invariant Gaussian random field (or Harrison-Zel'dovich spectrum) is very specific and has only two free parameters, the amplitude of the spectrum and the spectral index which measures the slight deviation from scale invariance predicted by inflation (perfect scale invariance corresponds to the idealized de Sitter universe).^[40] Inflation predicts that the observed perturbations should be in thermal equilibrium with each other (these are called adiabatic or isentropic perturbations). This structure for the perturbations has been confirmed by the WMAP satellite and other cosmic microwave background experiments,^[39] and galaxy surveys, especially the ongoing Sloan Digital Sky Survey.^[41] These experiments have shown that the one part in 10,000 inhomogeneities observed have exactly the form predicted by theory. Moreover, the slight deviation from scale invariance has been measured. The spectral index, n_s is equal to one for a scale-invariant spectrum. The simplest models of inflation predict that this quantity is between 0.92 and 0.98.^{[42][43][44][45]} The WMAP satellite has measured n_s = 0.960 ± 0.014^[46] and shown that it is different from one at the level of two standard deviations (2σ). This is considered an important confirmation of the theory of inflation.^[39]

A number of theories of inflation have been proposed that make radically different predictions, but they generally have much more fine tuning than is necessary.^[42][43] As a physical model, however, inflation is most valuable in that it robustly predicts the initial conditions of the universe based on only two adjustable parameters: the spectral index (that can only change in a small range) and the amplitude of the perturbations. Except in contrived models, this is true regardless of how inflation is realized in particle physics.

Occasionally, effects are observed that appear to contradict the simplest models of inflation. The first-year WMAP data suggested that the spectrum might not be nearly scale-invariant, but might instead have a slight curvature.^[47] However, the third-year data revealed that the effect was a statistical anomaly.^[39] Another effect has been remarked upon since the first cosmic microwave background satellite, the Cosmic Background Explorer: the amplitude of the quadrupole moment of the cosmic microwave background is unexpectedly low and the other low multipoles appear to be preferentially aligned with the ecliptic plane. Some have claimed that this is a signature of non-Gaussianity and thus contradicts the simplest models of inflation. Others have suggested that the effect may be due to other new physics, foreground contamination, or even publication bias.^[48]

An experimental program is underway to further test inflation with more precise measurements of the cosmic microwave background. In particular, high precision measurements of the so-called "B-modes" of the polarization of the background radiation will be evidence of the gravitational radiation produced by inflation, and they will also show whether the energy scale of inflation predicted by the simplest models (10¹⁵-10¹⁶ GeV) is correct.^[43][44] These measurements are expected to be performed by the Planck satellite, although it is unclear if the signal will be visible, or if contamination from foreground sources will interfere with these measurements.^[49] Other forthcoming measurements, such as those of 21 centimeter radiation (radiation emitted and absorbed from neutral hydrogen before the first stars turned on), may measure the power spectrum with even greater resolution than the cosmic microwave background and galaxy surveys, although it is not known if these measurements will be possible or if interference with radio sources on earth and in the galaxy will be too great.^[50]

As of 2006, it is unclear what relationship if any the period of cosmic inflation has to do with dark energy.^{[citation needed]} Dark energy is broadly similar to inflation, and is thought to be causing the expansion of the present-day universe to accelerate. However, the energy scale of dark energy is much lower, 10^-12 GeV, roughly 27 orders of magnitude less than the scale of inflation.

[edit] Theoretical status

In the early proposal of Guth, it was thought that the inflaton was the Higgs field, the field which explains the mass of the elementary particles.^[3] It is now known that the inflaton cannot be the Higgs field.^[51] Other models of inflation relied on the properties of grand unified theories.^[5] Since the simplest models of grand unification have failed, it is now thought by many physicists that inflation will be included in a supersymmetric theory like string theory or a supersymmetric grand unified theory. A promising suggestion is brane inflation. At present, however, whilst inflation is understood principally by its detailed predictions of the initial conditions for the hot early universe, the particle physics is largely ad hoc modelling. As such, despite the stringent observational tests inflation has passed, there are many open questions about the theory.

[edit] Fine-tuning problem

One of the most severe challenges for inflation arises from the need for fine tuning in inflationary theories. In new inflation, the slow-roll conditions must be satisfied for inflation to occur. The slow-roll conditions say that the inflaton potential must be flat (compared to the large vacuum energy) and that the inflaton particles must have a small mass.^[52] In order for the new inflation theory of Linde, Albrecht and Steinhardt to be successful, therefore, it seemed that the universe must have a scalar field with an especially flat potential and special initial conditions.

[edit] Andrei Linde

Andrei Linde proposed a theory known as chaotic inflation in which he suggested that the conditions for inflation are actually satisfied quite generically and inflation will occur in virtually any universe that begins in a chaotic, high energy state and has a scalar field with unbounded potential energy.^[53] However, in his model the inflaton field necessarily takes values larger than one Planck unit: for this reason, these are often called large field models and the competing new inflation models are called small field models. In this situation, the predictions of effective field theory are thought to be invalid, and renormalization should cause large corrections that could prevent inflation.^[54] This problem has not yet been resolved and some cosmologists argue that the small field models, in which inflation can occur at a much lower energy scale, are better models of inflation.^[55] While inflation depends on quantum field theory (and the semiclassical approximation to quantum gravity) in an important way, it has not been completely reconciled with these theories.

Robert Brandenberger has commented on fine-tuning in another situation.^[56] The amplitude of the primordial inhomogeneities produced in inflation is directly tied to the energy scale of inflation. There are strong suggestions that this scale is around 10¹⁶ GeV or 10^-3 times the Planck energy. The natural scale is naïvely the Planck scale so this small value could be seen as another form of fine-tuning (called a hierarchy problem): the energy density given by the scalar potential is down by 10^-12 compared to the Planck density. This is not usually considered to be a critical problem, however, because the scale of inflation corresponds naturally to the scale of gauge unification.

[edit] Eternal inflation

Main article: Chaotic inflation

Cosmic inflation seems to be eternal the way it is theorised. Although new inflation is classically rolling down the potential, quantum fluctuations can sometimes bring it back up to previous levels. These regions in which the inflaton fluctuates upwards expand much faster than regions in which the inflaton has a lower potential energy, and tend to dominate in terms of physical volume. This steady state, which first developed by Vilenkin,^[57] is called "eternal inflation". It has been shown that any inflationary theory with an unbounded potential is eternal.^[58] It is a popular belief among physicists that this steady state cannot continue forever into the past.^[59][60][61] The inflationary spacetime, which is similar to de Sitter space, is incomplete without a contracting region. However, unlike de Sitter space, fluctuations in a contracting inflationary space will collapse to form a gravitational singularity, a point where densities become infinite. Therefore, it is necessary to have a theory for the universe's initial conditions. Linde, however, believes inflation may be past eternal.^[62]

[edit] Initial conditions

Some physicists have tried to avoid the initial conditions problem by proposing models for an eternally inflating universe with no origin.^{[63][64][65][66]} These models propose that whilst the universe, on the largest scales, expands exponentially it is always spatially infinite and has existed, and will exist, forever.

Other proposals attempt to describe the ex nihilo creation of the universe based on quantum cosmology and the following inflation. Vilenkin put forth one such scenario.^[57] Hartle and Hawking offered the no-boundary proposal for the initial creation of the universe in which inflation comes about naturally.^[67]

Alan Guth has described the inflationary universe as the "ultimate free lunch":^[68] new universes, similar to our own, are continually produced in a vast inflating background. Gravitational interactions, in this case, circumvent (but do not violate) neither the first law of thermodynamics (energy conservation) nor the second law of thermodynamics (entropy and the arrow of time problem). However, while there is consensus that this solves the initial conditions problem, some have disputed this, as it is much more likely that the universe came about by a quantum fluctuation. Donald Page was an outspoken critic of inflation because of this anomaly.^[69] He stressed that the thermodynamic arrow of time necessitates low entropy initial conditions, which would be highly unlikely. According to them, rather than solving this problem, the inflation theory further aggravates it - the reheating at the end of the inflation era increases entropy, making it necessary for the initial state of the Universe to be even more orderly than in other Big Bang theories with no inflation phase.

Hawking and Page later found ambiguous results when they attempted to compute the probability of inflation in the Hartle-Hawking initial state.^[70] Other authors have argued that, since inflation is eternal, the probability doesn't matter as long as it is not precisely zero: once it starts, inflation perpetuates itself and quickly dominates the universe.^{[citation needed]} However, Albrecht and Lorenzo Sorbo have argued that the probability of an inflationary cosmos, consistent with today's observations, emerging by a random fluctuation from some pre-existent state, compared with a non-inflationary cosmos overwhelmingly favours the inflationary scenario, simply because the "seed" amount of non-gravitational energy required for the inflationary cosmos is so much less than any required for a non-inflationary alternative, which outweighs any entropic considerations.^[71]

Another problem that has occasionally been mentioned is the trans-Planckian problem or trans-Planckian effects.^[72] Since the energy scale of inflation and the Planck scale are relatively close, some of the quantum fluctuations which have made up the structure in our universe were smaller than the Planck length before inflation. Therefore, there ought to be corrections from Planck-scale physics, in particular the unknown quantum theory of gravity. There has been some disagreement about the magnitude of this effect: about whether it is just on the threshold of detectability or completely undetectable.^[73]

[edit] Reheating

The end of inflation is called reheating or thermalization because the large potential energy decays into particles and fills the universe with radiation. Because the nature of the inflaton is not known, this process is still poorly understood, although it is believed to take place through a parametric resonance.^[74][75]

[edit] Non-eternal inflation

Another kind of inflation, called hybrid inflation, is an extension of new inflation. It introduces additional scalar fields, so that while one of the scalar fields is responsible for normal slow roll inflation, another triggers the end of inflation: when inflation has continued for sufficiently long, it becomes favorable to the second field to decay into a much lower energy state.^[76] Unlike most other models of inflation, many versions of hybrid inflation are not eternal.^[77][78]

In hybrid inflation, one of the scalar fields is responsible for most of the energy density (thus determining the rate of expansion), while the other is responsible for the slow roll (thus determining the period of inflation and its termination). Thus fluctuations in the former inflaton would not affect inflation termination, while fluctuations in the latter would not affect the rate of expansion. Therefore hybrid inflation is not eternal. When the second (slow-rolling) inflaton reaches the bottom of its potential, it changes the location of the minimum of the first inflaton's potential, which leads to a fast roll of the inflaton down its potential, leading to termination of inflation.

[edit] Inflation and string cosmology

The discovery of flux compactifications have opened the way for reconciling inflation and string theory.^[79] A new theory, called brane inflation suggests that inflation arises from the motion of D-branes^[80] in the compactified geometry, usually towards a stack of anti-D-branes. This theory, governed by the Dirac-Born-Infeld action, is very different from ordinary inflation. The dynamics are not completely understood. It appears that special conditions are necessary since inflation occurs in tunneling between two vacua in the string landscape. The process of tunneling between two vacua is a form of old inflation, but new inflation must then occur by some other mechanism.

[edit] Inflation and loop quantum gravity

When investigating the effects the theory of loop quantum gravity would have on cosmology, a loop quantum cosmology model has evolved that provides a possible mechanism for cosmic inflation. Loop quantum gravity assumes a quantified spacetime. If the energy density is larger than can be held by the quantified spacetime, it is thought to bounce back.

[edit] Alternatives to inflation

String theory requires that, in addition to the three spatial dimensions we observe, there exist additional dimensions that are curled up or compactified (see also Kaluza-Klein theory). Extra dimensions appear as a frequent component of supergravity models and other approaches to quantum gravity. This raises the question of why four space-time dimensions became large and the rest became unobservably small. An attempt to address this question, called string gas cosmology, was proposed by Robert Brandenberger and Cumrun Vafa.^[81] This model focuses on the dynamics of the early universe considered as a hot gas of strings. Brandenberger and Vafa show that a dimension of spacetime can only expand if the strings that wind around it can efficiently annihilate each other. Each string is a one-dimensional object, and the largest number of dimensions in which two strings will generically intersect (and, presumably, annihilate) is three. Therefore, one argues that the most likely number of non-compact (large) spatial dimensions is three. Current work on this model centers on whether it can succeed in stabilizing the size of the compactified dimensions and produce the correct spectrum of primordial density perturbations. For a recent review, see^[82][83]

The ekpyrotic and cyclic models are also considered competitors to inflation. These models solve the horizon problem through an expanding epoch well before the Big Bang, and then generate the required spectrum of primordial density perturbations during a contracting phase leading to a Big Crunch. The universe passes through the Big Crunch and emerges in a hot Big Bang phase. In this sense they are reminiscent of the oscillatory universe proposed by Richard Chace Tolman: however in Tolman's model the total age of the universe is necessarily finite, while in these models this is not necessarily so. Whether the correct spectrum of density fluctuations can be produced, and whether the universe can successfully navigate the Big Bang/Big Crunch transition, remains a topic of controversy and current research.

[edit] See also

• Brane cosmology
• Varying speed of light
• Dark flow

The gravitational effects produced by a given mass are described in general relativity by 16 coupled hyperbolic-elliptic nonlinear partial differential equations, called the Einstein field equations. As result of the symmetry of  and , the actual number of equations reduces to 10, although there are an additional four differential identities (the Bianchi identities ) satisfied by , one for each coordinate.

The nonlinearity of the Einstein field equations stems from the fact that masses affect the very geometry of the space in which they dwell. And this is the fundamental insight of : mass curves the geometry of spacetime, and the geometry of spacetime in turn tells masses how to move.

Brans-Dicke Theory, Cosmological Constant, Cosmological Equations, General Relativity (Weisskopf website).

The Einstein field equations (EFE) may be written in the form: (Wikipedia website).

 

where R_μν is the Ricci curvature tensor, R the scalar curvature, g_μν the metric tensor, is the cosmological constant, G is the gravitational constant, c the speed of light, and T_μν the stress-energy tensor.

This then is the Einstein Field Equation underpinning all of cosmology.

The Einstein-Riemann Tensor Guv=Ruv-½guvR = -8πGTuv/c⁴ relates the Riemann-Metric guv with scalar tensor R to the Ricci-Tensor Ruv for a stress-energy density tensor Tuv. The Weyl Curvature in Ruv preserves volume as a tidal shear effect, whilst the Ricci Curvature acts on the density and changes the density and so the volumes.

The Weyl Curvature Nullification hypothesis of Roger Penrose (Oxford University, UK) shows, that the Weyl Curvature must become 0 at the threshold between General Relativity's metrics and the 'singularity' of quantum mechanics for the selfconsistency of the physical universe to hold in its inertial parameters.

A 0 Weyl curvature means that the Lorentz Contraction of a tangential displacement vector travelling around a 'wormhole singularity' or Weyl-Centre as Black Hole event horizon must dewarp itself at that wormhole perimeter in accompanying invariance of the scalar orthogonal radius vector not subject to the Lorentz contraction of Special Relativity in say a rotating system.

We shall describe this Weyl-Limit as a superbrane parameter negating the mathematical singularity of General Relativity in a minimum superstring condition:  λ_ps=2πr_ps.

This then, if written out in Newtonian forms give the FRLW-Cosmology and the dynamics as described below (Wolfram website).

Astrophysics Cosmology Friedmann-Lemaître Cosmological Model      A cosmological model formulated by Friedmann in 1922 and independently by Lemaître (1927). It assumes a homogeneous and isotropic universe with positive cosmological constant and expansion parameter a governed by the equations of motion     (1) and (2) where is the total mass-energy density, is the mass density, and c is the speed of light. These equations were originally rigorously derived from general relativity, but can also be rigorously derived from Newtonian mechanics and thermodynamics for small length scales, i.e., r such that

 

In the Friedmann-Lemaitre cosmological model the universe is homogenous and isotropic, that is it doesn't change in its uniform composition in any directional sense. Here the cosmological constant Lambda (Λ) is taken as positive for various curvatures k and a nonzero pressure P.

In the Einstein-de Sitter cosmological model this universe has also constant curvature k, but for a Λ=0=P.
 

The equations of motion for the two models depend on a scale factor R(n)=a(t).R_o, meaning that the expansion of  comoving coordinates becomes relative to an arbitrarily fixed scale R=R_o at a time t=t_o.and for the scale factor a(t) being dimensionless.
In the contemporary cosmologies, R_o is taken to be the present scale of the universe at a scale factor a(t_o)=a_o and for a 'present' Hubble-Constant H_o, but we shall find, that it is more appropriate to take the R_o scale to become a 'relatively' fixed Hubble-Horizon  R_Hubble=c/H_o so describing not a present epoch dependent Hubble function H(t), but  a 'Nodal Constancy' set 'fixed' at the beginning of the universe in its de Broglie Inflation.


 

The Newtonian Cosmology derives the FRLW-Equation of motion from Energy Conservation.

Consider the Universe of total mass M to be a spherical distribution of 'pointmasses' m; that is this total mass M is distributed within a typical 3-D volume radius r.

Newton's Law for Gravitation:   F=GMm/r²

A 'point mass' m so becomes gravitationally 'attracted' to the large mass M and with r the separation between M and m and F is in the direction r or F = GMm r/r³  in (bold) vector notation.

Integrating from r to infinity ∞, we obtain the gravitational potential energy V=-GMm/r=∫GMm/r² dr.

Now let a spherical volume of radius r (~100Mpc~326 million lightyears as the typical supercluster-void scale) be sufficiently large, so that space can be considered homogeneous and the energy interaction between the pointmass m and the 'total mass-seed' M describes their interacting dynamics.

The Energy Equation so is  U=T+V  or Total Energy=Kinetic Energy + Potential Energy.

The total energy U of a test particle m at r will be the sum of the potential V=-GmM/r and kinetic energy  T=½mv²  and for  velocity:  (v=dr/dt) and acceleration:  (A=dv/dt=d²r/dt²=dA/dv.dv/dt=d[½v²]/dr).

Mass M can be written by Density=ρ=Mass/Volume as: M=ρV for the differential dM=ρdV=4πρr²dr  for a spherical volume V=4πr³/3.

U=T+V = ½mv²  -GMm/r =  ½mv²  -GρVm/r = ½mv²  -4πρGr²m/3.

This then is the evolution of the separation of r from the origin or, invoking the cosmological principle of observer independence; the separation r between any two particles as description for the expansion of the Newtonian universe.

The CRITICAL boundary condition for U becomes U=0 for v=0 at an infinite radius r=∞.

If U=0, then the universe is perfectly balanced between its kinetic expansion and its gravitational contraction. The universe then is INFINITE is space and is Euclidean FLAT with a Curvature k=0.
If U<0, then the universe is dominated by the gravitational contraction and becomes CLOSED in a positive curvature of k=+1 in a FINITE spherical space.
If U>0, then the universe is dominated by the thermo-kinematic expansion and is OPEN in a negative curvature k=-1 in INFINITE hyperbolic space.
 
For a uniform expansion,  the relationship between r=R and a comoving distance x=R_o can be written in the scalefactor a(t) as: R(t)=a(t).R_o and for homogeneity a(R,t)=a(t) for all displacements R and with dR/dt=(da/dt)R_o.

The coordinate grid or metric expands with time as the universe expands, but local inertial reference frames remain at fixed locations in the comoving system as a comoving reference frame.

So it is a(t) as the scale factor which determines how physical separations change with time.
If a(t) doubles, the separation of all local inertial frames doubles.

Using the scale factor, as R(t)=a(t).R_o,  and v=dR/dt=R_o.da/dt,  the total energy becomes:

U = ½m(R_o.da/dt)²  -4πρGR²m/3 =  ½m(R_o.da/dt)²  -4πρmG[a(t).R_o]²/3

As the total energy U per unit mass m must be conserved over any time t,
dU/dt=0 for all times t:

U/m=c² = ½v²  - 4πρGr²/3 
dU/dt=d(c² )/dt=0=d(½v²)/dt - d(4πρ(t)Gr²/3)/dt  for
dU= d(½v²) - d(4πρ(t)Gr²/3)=0 and

The factor 2U/m(cR_o)²  =k is called the CURVATURE k with Curvature Radius 1/R_o² for
2U/mR_o² = kc² = (da/dt)² - 8πρ(t)G(a)²/3 =½(R_o.da/dt)² -  4πρ(t)G(a.R_o)²/3  

2U/m(aR_o)² = kc²/a² =(da/dt)²/a² - 8πρ(t)G/3 

{(da/dt)/a}² = 8πρ(t)G/3 - k(c/a)²  = 8πρ(t)G/3 - (c/aR_o)²     {Equation #1}
 or  (da/dt)²  = 8πρ(t)Ga²/3 - kc²  
 
 (da/dt)² =(dR/dt)²/R_o² = 8πρ(t)G(R/R_o)²/3 - kc² (+Λ/3)

This is the Friedmann-Lemaitre Equation from above for U= ½mc²  for the kinetic energy T=½mv²  and without a cosmological constant (Λ/3).

To preserve homogeneity, k must be independent of R_o, so while U is constant for a given particle m it changes with separation R_oas U↔(aR_o)².

The curvature k retains its value throught its evolution in the Friedmann universe.

In terms of acceleration A=dv/dt=d²r/dt²=d[½v²]/dr=v.dv/dr = d(½[dr/dt]²)/dr then
 A=d²R/dt²= d(½[dR/dt]²)/dR =d(4πρ(t)GR²/3 - ½c²  + Λ/6)/dR
Integration yields:  ½[dR/dt]² = 4πρ(t)GR²/3 - ½c² + ΛR_o²/6

To include this cosmological constant and for the purpose to obtain v=dR/dt=0, we incorporate Constant =-ΛR_o²/3 = 8πρ_oGR_o²/3 - c²  and set Λ/3 = 8πρ_oGa_o²r_o²/3 -2U/mc�x�. This is the

Critical DensityNow consider a sphere of comoving radius R (expanding along with the universe) enclosing a mass M. The "Energy equation" U/m=v�/2-GM/R applied to a test mass m tells us for U=0 v=0 at R=infinity and the system is "critical" or just unbound.

Mcrit=v�R/2Gor, with M=rV=4/3prR� we can write the mass density as M/V: 3Mcrit/4pR�=3v�/8pGR� and in an expanding universe, v=HR on the comoving sphere so 3Mcrit/4pR�=3H�/8pG. The quantity 3H�/8pG does not depend on R ... This quantity is called the critical density of the universe... rcrit=3H�/8pGIf the average density of the Universe is less than or equal to critical it will expand forever, and the Universe is open. If greater, the Universe is closed. A Universe with critical density is flat, infinite, and the expansion rate approaches zero at t=infinity. If H=71 km/s/Mpc then rcrit=9.45E-27 kg/m� (about E-26 kg/m� or E-29 gm/cm�).
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The Cosmological Constant LRewrite the energy equation as v�=2GM/R+2U/m.Suppose the Energy U is related to m via U~mc�... so U/m=-kc� or-kc�/2 (choice depends on whether you like k to be 0,�1 or 0,��) ...then v�=2GM/R-kc�.For a flat Universe k=0, k=+1 for a closed and -1 for an open Universe. (In GR other values of k are not allowed.) Note that for k=+1 if R>2GM/c� v� goes negative, not allowed... this is a closed universe.
Now we can put the cosmological constant Lambda into our energy equation:

v�=2GM/R-kc�+LR�.
The effect of the new term is to make the expansion rate increase exponentially with time. The new term dominates when

L>2GM/R� or L>8prG/3.
and R(t)=RoeHt/to Useful for inflation in the early Universe or the new results from SNeIa and WMAP etc.
Einstein introduced the constant to keep the universe from expanding (biggest blunder?) by setting k=0 and L=-8prG/3 which results in v=0. Of course Einstein used real relativity! He added a term L to his field equation: Gmn=Tmn+gmnL Gamow in My World Line 1970 (Viking, New York) writes "when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder he ever made in his life."
The latest results imply L=0.73 and the matter density 0.27 (total is unity) which gives an older Universe than if L were zero, since the expansion was slower in the past, relieving the "Universe is too young" problem.

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Now use the first law of thermodynamics dE + p dV = T ds
applied to E = mc� = 4/3pa�rc�
Then a change of E in time dt is

dE/dt = 4pa�rc�(da/dt) + 4/3pa�(dr/dt)c�
while the volume changes by

dV/dt = 4pa� da/dt
and if dS=0 (reversible) the first law gives the fluid equation

{dE/dt + dV/dt = 0} times 3/4pa�c�   or dr/dt + 3(da/dt)/a (r+p/c�)=0. 

The density evolution is due to volume increase (adot/a times r) and loss of energy due to work done as the volume of the universe is increased (adot/a times p/c�) = the energy converted into gravitational potential.

Now if we only knew how P varies, the equation of state ... P=P(r,T??)

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If we differentiate (d/dt) the Friedmann equation we get

d/dt[(da/dt)/a]�=2[(da/dt)/a] [a(d�a/dt�)-(da/dt)�]/a� = 8/3pG(dr/dt) + 2kc�(da/dt)/a�
and substituting for rhodot from the field equation we have

(d�a/dt�)/a - [(da/dt)/a]� = -4pG(r+p/c�) + kc�/a�
so from the Friedmann equation again we see

(d�a/dt�)/a = -4/3pG(r+3p/c�)
and yes, that is +3p/c�, any pressure actually decreases (decelerates) the expansion!
(The curvature k cancelled out.)

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Most GR equations use "natural units" where c=1 so the Friedmann equation becomes

[(da/dt)/a]� = (8pG/3)r - k/a�
then k has units t-2 (and time is measured in metres).

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The Hubble LawHubble observed v=Hr where since r=a(t)x
v=[(da/dt)/a]dr/|r| so in the spirit of the Friedmann equation
H=(da/dt)/a.
Since H=H(t) we denote the value of H we observe today Ho and, since Ho>0 we know the universe is expanding. (This is in terms of our Friedmann universe, but the Hubble expansion can be explained in simpler ways and usually is.) But using H� for (adot on a) squared we have

H�=(8pG/3)r - k/a�    (c�=1)
So H=H(t) and H should decrease with time as the expansion of the universe is slowed by gravitational attraction. If the Friedmann model is right and if there is no extra "dark physics".

The redshift used to justify the big bang is a result of the time dependent scale factor adot on a... the time between emission and absorption of a photon is dt=dr/c,

dl/lo=[(da/dt)/a][dr/c]=[(da/dt)/a]dt=da/a
where dl=l-lo, lo is the "lab" wavelength, l is the (redshifted) wavelength, and integrating:

ln(l/lo)=ln(a) i.e. lµa.
We usually talk in terms of a "redshift parameter" z=Dl/lo (if you don't see a subscript it is there!) then "redshift" z is related to the scale of the universe by

1+z=l/lo=ao/a.
So if the wavelength has doubled, z=1 and the universe is twice as large as when the light was emitted. (The universe was half the present age when the light was emitted.)

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The Equation of State for -- DustDust? Well, what we mean here is any material that exerts negligible pressure so we can set p=0. This is obviously the simplest case but it is not trivial since it is nearly exact for stars and galaxies which really only interact gravitationally (no star collisions even when galaxies collide!) and not too bad for a universe that has cooled to the point where atoms are neutral and collisions infrequent. Good for dust too, but there is not too much of that in the universe. We use the term "dust" for nonrelativistic matter like the stellar interior people use "z" for "metals", anything not H or He!
The Equation of State for radiationRelativistic particles, photons included, exert a pressure p=rc�/3. For photons this is called radiation pressure. Neutrinos may also be important? (Neutrinos have to be included if they have rest masses of order of an eV.) Now we can solve a couple of interesting cases....
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Friedmann Universe for k=0 and p=0The fluid equation

dr/dt + 3(da/dt)/a (r+p/c�)=0
for dust becomes dr/dt + 3r(da/dt)/a=0

i.e. d(ra�)/dt=0   (zero times a� if you wonder where that factor went!)

Integrating, ra�=constant. Hardly surprising, the density goes as one over the volume! But this is useful since as we are only interested in a ratio of adot to a we can rescale a(t) and it is usual to choose a(present)=1. Then the physical and comoving coordinate frames coincide at the present time and we can write

r=ro/a�
and if we substitute this into the Friedman equation with k=0 we get

[(da/dt)/a)]� = [8pG/3] ro/a�

[da/dt]� = [8pGro/3] [1/a]
which obviously :) has a solution a µ tq... hmmmm.... the LHS would be adot squared goes as t to the minus 2q-2 and the RHS 1/a goes as t to the minus q so setting the powers equal we have 2q-2=-q or q=2/3. (Wow!) So a(t) goes as t2/3 and since (we set now=to) a(t)=[t/to]2/3 and

r(t) = ro/a� = ro(to/t)�.
The universe expands forever but the expansion rate decreases with time: H goes as 2 over 3t

[(da/dt)/a] = H = 2/3t.

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Friedmann Universe for RadiationIf the equation of state is p=rc�/3 the fluid equation

dr/dt + 3[(da/dt)/a] (r+p/c�)=0   becomes

dr/dt+4[(da/dt)/a]r = 0.
Solving this as before we find now r=ro/a4   instead of /a�. Applying this to the Friedmann equation and guessing a solution :) we get a=a(t)=(t/to½) so

r(t) = roa(t)-4 = ro(to/t)2.
The expansion is slower if radiation dominated (extra deceleration due to pressure) but as before density falls off as t�. The density falls as the fourth power of the scale factor because of increasing volume (three of the four) and redshift of the radiation (the other one.) The drop can also be thought of as the work done p dV by the pressure as the univeerse expands.

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Mixtures of Dust and Radiation
The total density is rtot = rdust + rrad   where rdustµa-3   and rradµa-4.

Usually one or the other will dominate, but if radiation dominates note that the rapid fall with scale will eventually cause the "dust" to become important and after a long enough time dominate. At present dust dominates and so will continue to do so for the rest of time.

.

Ho=c/Rmax in demetrication}.

Ho=c/Rmax in demetrication}.
So r(dot)^2=(a(dot).ro)^2=2GM(r)/r + L(aro)^2/3 - c^2.

This becomes: {a(dot)/a}^2=2GM(r)/r(aro)2 - c^2/(aro)^2 + L/3=8pGr/3-c^2/(aro)^2+L/3............. (FLM1)

Now consider the universe's expansion to be adiabatic, that is thermodynamically closed. Energy E and the pressure (P) variation with respect to Volume V sum to 0 change in the 'heat content Q' (or enthalpy H=U+PV for internal heat content U).

dQ=dE+PdV=0; dE/dt+PdV/dt=0 for E=M(r)c^2=4pSrR^3c^2/3 and total density Sr=rmatter +pressure

d{SrR^3}/dt=-(3P.R^2/c^2).dR/dt =R^3.r(dot)+3R^2.r.R(dot) for

r(dot)+3r.(a(dot)/a)=-(3P/c^2).(a(dot)/a) and the dynamical equation:

r(dot)+3(a(dot)/a){r+P/c^2}=0....... (FLM2)

The combined Friemann-Lemaitre equation of motion for matter density r then is:

{a(dot)/a}^2 = 8pG/3{r+3P/c^2} - c^2/(aRo)^2 + L/3..........................................................(1)

The Equation of motion in the Einstein-de Sitter cosmology then sets P=L=0 and a constant curvature k=1/Ro^2=0 in the Friedmann-Lemaitre model for:

(a(dot)/a)^2=8pGr/3 = H^2 with R=aRo=aRmax

Solving for a(dot)^2=2GM/aRmax^3 via Sqrt(a).da=Sqrt(2GM/Rmax^3)dt leads to

a.Rmax=Cuberoot{9GM/2}.t^[2/3] for a limiting boundary condition ao=0 for to=0; (which we shall see is actually n=nps=Ho.tps for a=1/(1+Rmax/lps).

Using H=a(dot)/a, 9GM/2=(aRmax)^3/t^2 for H=Sqrt(4/9t^2)=(2/3t) and the Hubble Time becomes

1/H=3t/2 as the age of the universe for time t.

We shall show that this Hubble Time actually represents the completion of a Hubble-Oscillation and so the LIGHTPATH R(n)=ct where Rmax necessarily represents a semiwavelength as the distance between the even nodes 0,2,4,6.. and the odd nodes 1,3,5,7,....

For define R(n)=Rmax(n/(n+1)) with n=Hot and a=(n/(n+1)) for a lightpath of 2Rmax =ct*=2c/Ho, then this time t* given in c-invariance in say 11D/5D hyperspace will be reduced in n=2 for R(2)=Rmax(2/3)=2c/3Ho for the matter dominated cosmology in 10D/4D.

This then maps the 'nodally corrected' Hubble Law as Ho=2c/3R(2) onto the old H=2/3t.

The Friedmann-Lemaitre cosmology, incorporating the pressure and lambda terms solves in terms of the curvature k=1/Ro^2. We use R=aRo and R(dot)=a(dot)Ro and a(dot)/a=R(dot)/R=H and the definition of Omega (W)=r/rcritical=8pGr/3Ho^2 andwrite (1) as a COSMOLOGICAL EQUATION:

R(dot)^2=8pGrR^2/3+LR^2/3-kc^2 .

So curvature kc^2=R^2{Ho^2W+L/3-Ho^2}=(aRo)^2{Ho^2(W-1)+L/3}..............(Curvature*)

Then for (R/c)^2=(aRo/c)^2=(R(n)/c)^2=(Rmax(n/(n+1))/c)^2=(a/Ho)^2, k=0 iff (a/Ho)^2{Ho^2(W-1)+L/3}=0

which is the case for W=1 and L=0 OR for L(W-1)=-3Ho^2 (as frequency squared cycle units).

In the demetricated scenario W=0.028 with a varying L as quintessence in the mass parametric and open cosmology, however encompassed by W=1 in the electromagnetic oscillatory closure.

Then as Ho=1.877728045x10^-18 1/s* , 'L'=constant=3Ho^2/0.972=1.0882292x10^-35 and of the order of the Planck Scale.

This indeed is the experimental observation of the 'cosmological constancy'.

The curvature is 1 for 'L'=3Ho^2{2-W+2/n+1/n^2} and a function of cyclenumber n however.

Then for the initial condition of the inflaton and the instanton, n=nps=lps/Rmax=6.259..x10^-49 and 'L' is upper bounded in 3c^2/lps^2=2.7x10^61 frequency units.

This becomes simply 3fps^2 as the source frequency in the demetrication for the constant 'L', expressed in the quintessence of the variable L in the de Broglie inflaton of the 0-node discussed later.

For n=1 and the first odd node at the semi-wavelength for the Hubble Oscillation 'L'=5.26x10^-35 frequency units for the 0 approximation.

At the completion of the Hubble Oscillation n=2 and 'L'=3.41x10^-35 frequency units and decreasing towards 0 after infinite time t=n/Ho.

This then solves the cosmological constant dilemma in the superpositioning of spacetimes.

Next we define 1+z=a/ao specifying the deceleration parameter q.

q=-[a(doubledot)/a]/[a(dot)/a]^2=-(a(doubledot)a)/(a(dot))^2n=-(a(doubledot)/a)/H^2 in say Taylor-Expansion a(t) about t=to, that is some initial time to where a=ao and H=Ho, which in the demetrication is a double value Rmin=lps=c/fps,Ro=Rmax=c/Ho for k=0,1 for no=nps,infinity limit and ao=no/(no+1) or

lps/Rmax and 1 respectively.

a(t)=a(to)+a(dot)(to)[t-to]+(a(doubledot)(to)/2)[t-to]^2+...=ao{1+Ho[t-to]-(qo/2)Ho^2[t-to]^2+...}.

So 1+z=1+Ho[t-to]+(-qo/2)Ho^2[t-to]^2+...=a/ao=fo/f=Ho/H.

Then density ro=(a/ao)^3.W.rcritical=(a/ao)^3.W.3H^2/8pG...........(boundary density).

This becomes (1+z)^2={(1+v/c)/(1-v/c)} in demetrication with v=c/(1+n)^2=R(dot).

Then (1+z)^2=(n^2+2n+2)/(n^2+2n)=1+2/{(n+1)^2-1} and

a/ao=1+z=Sqrt{1 + 2/[(n+1)^2-1]}=Sqrt{1+2/[(c/v)-1]}..............................Expansion-Redshift-Parameter